Question

Which of the following is true?

(a) WN*=\(\frac{1}{N} W_N^{-1}\)

(b) WN^-1=\(\frac{1}{N} W_N*\)

(c) WN^-1=WN*

(d) None of the mentioned

I got this question in a job interview.

This is a very interesting question from Frequency Domain Sampling DFT topic in division Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

Correct answer is (b) WN^-1=\(\frac{1}{N} W_N*\)

The explanation: If XN represents the N point DFT of the sequence xN in the matrix form, then we know that

XN=WN.xN

By pre-multiplying both sides by WN^-1, we get

xN=WN^-1.XN

But we know that the inverse DFT of XN is defined as

xN=\(\frac{1}{N} W_N*X_N\)

Thus by comparing the above two equations we get

WN^-1 = \(\frac{1}{N} W_N*\)